The centered trace and its comparison exclusions

The centered trace and its comparison exclusions

A selected observable and an all-distance conclusion. The centered trace belongs to the selected theory on its common operator domain. Its two-point function is represented by positive spectral measures patched from the same physical hierarchy and phase. Universal selection along arbitrary radii, combined with monotonicity, gives the full short-distance limit. This is stronger than observing the value on one favorable scaling sequence and requires the station-selection argument just proved.

The measure, Laplace and Schwartz passages below retain their polynomial bounds and test classes. They recover log) = 3 dim g lim

The positive constant describes an ultraviolet asymptotic. It is not a positive mass-gap bound. The ensuing finite mass-shell obstruction uses the precise finite polynomial comparison class, while the Maxwell arguments use their own coherent-sector and locality hypotheses. Neither exclusion is silently enlarged to every conceivable free comparison.

24.1 The centered physical trace and its full short-distance limit

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In the selected character vacuum use the same conserved physical stress tensor and its degree-four
Euclidean trace T = P3
μ=0 Tμμ. Put Θ = T −⟨T⟩1. On the common field domain,

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⟨Ω, Θ(f)Ω⟩= 0,
Θ(f) = T(f) −⟨T⟩
Z
f

I.
(TR1)

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Indeed the Euclidean unit law removes a unit slot. Strict complex-time functions agree on imaginary
times and then by holomorphic uniqueness. Fubini smearing gives the factor
R f, and tempered
boundary passage gives the Wightman identity. Pairing with the dense word span makes unit in-
sertion exactly (
R f)I on the domain. Linearity gives TR1; the zero Euclidean one-point function
passes through the same continuation and boundary. Reality and scalar covariance give a Her-
mitian Lorentz-scalar pinned field. For s > 0 define E(s) = ⟨Θ(−se0/2)Θ(se0/2)⟩E. The source
normalization gives

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lim
s↓0 s8 log2(1/s)E(s) = AG > 0,
AG = 12d2(G)2
b0(G)2
dim g
π4
,
d2(G) = −b0(G)/2,
AG = 3 dim g
π4
.
(TR2)

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The nonabelian simple algebra has positive dimension and b0 > 0, so the coefficient is positive.
These are the physical-stress and Wilson-calibrated running-charge conventions. Translation and
orthogonal invariance reduce every separated trace pair to E at its Euclidean distance. An orthog-
onal reflection takes their difference to a time-axis vector of the same norm; a common translation
then puts them at the opposed points above. Degree-four scaling identifies the rescaled trace with
4d2 times the free scalar label after logarithmic running normalization. The scalar Wick contraction
is 3 dim g/(π4|x0 −x1|8). Multiplying by running factor b−2
0 /4 and trace factor (4d2)2 gives exactly
AG|x0 −x1|−8 on the extracted radii and nearby generic plots. The positive spectral representation

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E(s) =
Z
V +
e−sp0 dμΘ(p)
(TR3)

makes real, nonnegative and nonincreasing. Choose generic distances arbitrarily near one; their scaling limits are and and their nearby actual distances bracket the exact radius. Monotonicity sandwiches its normalized value; let1. If the all-distance TR2 limit failed, choose violating distances tending to zero. The scaling construction admits a refining subsequence of these very distances with compatible moving plots and degree-four comparison. The same sandwich contradicts the violations. This proves the full positive-distance limit, beyond the earlier weak-sequence comparison.

Centered-field and all-distance recovery. Let be the reconstructed Hilbert-space unitary to its horizontal realization, with Ω Ωand common field domain . Retain the selected character under the ground-algebra isomorphism and set and 1 1. Their scalar vacuum readings give

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⟨T Z⟩= ⟨T⟩,
ΘZ = J4(T −⟨T⟩1),
ΘZ(f)V ψ = V Θ(f)ψ
(ψ ∈D).

Unit removal through the same strict-time continuation, smearing and tempered boundary gives, on this actual domain,

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1Z(f)V ψ =
Z
f

V ψ,
ΘZ(f) = T Z(f) −⟨T⟩
Z
f

IZ.

Here) denotes the reconstructed field of the lifted label. These identities recover TR1, its zero vacuum expectation, Hermiticity and scalar covariance. They assert operator equality on the common domain; no bounded-field extension is inserted. To check the identity operator in this formula, first insert a unit label in any Euclidean correlation and use the hierarchy’s unit law to remove it.

The separated Euclidean readout is unchanged, so). Its amplitude retains all normalization factors separately:

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3 dim g
π4
1
4b2
0
(4d2)2 = 12d2
2
b2
0
dim g
π4
= 3 dim g
π4
= AG > 0.

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This uses exactly d2 = −b0/2 and the physical stress/running-charge conventions, which are re-
tained rather than independently rescaled. The transported positive spectral measure gives the
same Laplace reader; since p0 ≥0, for s2 ≥s1 > 0 it satisfies 0 ≤EZ(s2) ≤EZ(s1).
To expose the all-distance quantifiers, let rj ↓0 be any proposed violating array and take
its source-permitted refining scaling subsequence. For fixed generic a < 1 < b near one, retain
the actual nearby distances d−
j < rj < d+
j supplied by the moving plots, with the corresponding
normalized limits. Put N(r) = r8 log2(1/r) > 0 for sufficiently small r. Exactly

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N(rj)EZ(d+
j ) ≤N(rj)EZ(rj) ≤N(rj)EZ(d−
j ).

The outside limits are and from the retained scaling comparison. Taking to one forces the middle limit to, contradicting a fixed positive violation tolerance. All distances, refinements and scalar readings are identical after recovery. Thus the lift retains the full positive-distance limit TR2 and its dependence on the positive spectral representation TR3, without replacing the earlier weak-sequence statement by itself.

The unique positive trace measure has the reconstruction normalization

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∥Θ(f)Ω∥2 =
Z
| bf+(p)|2dμΘ(p),
⟨Θ(f)Ω, Θ(g)Ω⟩=
Z
bf+(p)bg+(p)dμΘ(p).
(TR4)

It has forward-cone support, Lorentz invariance and temperate growth; every positive time damping has finite mass. Its relative Fourier transform gives the full two-point distribution on joint Schwartz tests. The joint momentum measures on dense smeared trace vectors patch to this positive measure; polarization and compact-test separation give uniqueness. The regularized point-vector construction TR5 and the measure argument TR10–TR17 below give the patching and the ultraviolet limit explicitly. The lifted trace source retains the selected vacuum, stress label, unit insertion and actual common domain. Centering readout recovers TR1 exactly. The spectral reader retains its positive measure and monotonicity, so the generic scaling reader extends to all positive distances without selecting a new theory. The constants in TR2 retain both the running and trace factors, recovering the stated logarithmic coefficient; this asserts no positive mass gap.

24.2 The full short-distance limit: monotonicity, universal radius selection and the sandwich

It is real and nonnegative. Since0 almost everywhere for this measure, 0 implies , and integration proves). Finiteness at every positive is part of the positive-time spectral representation. Thus this monotonicity is available before proving the short-distance limit; it is not inferred from that limit.

We give the precise selection statement needed here. Fix the physical table, its phase decomposition and its stress tensor. The trace construction selects a comparison ultrafilter below the table’s mesh-then-volume filter, its retained local data, and one free hierarchy. For every sequence

tending to zero, it then provides a strictly increasing map, a scaling family on this same physical table, and its comparison data, whose radii satisfy the exact identity

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ri = sj(i).
(TLim1)

The scaling comparison applies to these radii. In particular the statement is not merely that one favorable sequence0 exists. Nor does it select a different physical hierarchy or phase when the sequence changes. The construction and this universal sequence property were established in the preceding interacting-to-Gaussian comparison. First prove convergence along the radii of any such retained comparison. For 1 2, put

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x(u)
0
= 0,
x(u)
1
= (u/2, u/2, u/2, u/2).

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The four coordinates of the two points are distinct, so this is a generic two-point configuration,
and its Euclidean separation is u.
Its maximum-coordinate separation is u/2 > 1/4, while its
configuration norm is less than 2. Consequently it has a neighborhood in the fixed comparison
window with norm bound 100 and maximum-coordinate separation at least 1/100.
Let bx(u)
i
be the rounded configurations in that comparison, and set di(u) = |bx(u)
i,0 −bx(u)
i,1 |. Round-

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ing convergence gives di(u) →u. The actual physical configuration is exactly ribx(u)
i
, not just an
asymptotic replacement of it. The interacting comparison, followed by the running-clock identity
and the free scalar Wick contraction, gives

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c(ri)E(ridi(u)) −→AGu−8,
c(r) = r8 log2(1/r).
(TLim2)

The centering and scalar homogeneity used in this identity are retained in the trace-label calculation below. The same normalization), independent of, is used on both sides of the coming inequalities. Fix 2). Eventually 0). Monotonicity and0 give

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c(ri)E(ridi(b)) ≤c(ri)E(ri) ≤c(ri)E(ridi(a)).

By (TLim2), their outer limits are and. Given 0, first choose close enough to 1 that both outer limits differ from by less than2. Then choose large enough for the two convergences and the bracketing inequalities. This proves. The complex pairing has zero imaginary part by (TR3), so this also proves complex convergence.

Finally, suppose the full right-hand limit at zero failed. There would be 0 and, for every a radius 0 such that

Apply the universal selection statement to this particular sequence. Equation (TLim1) retains this inequality at every, whereas the preceding sandwich proves that its left side tends to zero. This contradiction proves (TR2) for the full small-scale limit. No density or ratio condition on a single favorable sequence of radii is being assumed.

Source–readout formulation. The reader) of the selected phase is recovered at all small, although each comparison witness (a scaling family) covers only a sequence of radii. The witnesses form a cover of every sequence, and the recovered quantity is independent of the witness: the scaling comparison supplies, for an arbitrary sequence tending to zero, a subsequence along which the limit is, on the same table and phase, and the sandwich by monotonicity then forces the common value. No density or ratio condition on a favourable sequence of radii is assumed. Monotonicity of is available before, and independently of, the short-distance limit, because it is read from the positive-time spectral measure.

Theorem 24.1 (Full limit from a universal subsequence property). lim

Proof. Fix 2). Eventually 0), and by monotonicity and

with outer limits and. Choosing close to one and then large gives along every retained comparison. If the full limit failed there would be 0 and radii with; applying the universal selection to this sequence, (TLim1) keeps the inequality at every, contradicting the convergence just proved.

With = 3 dim g from the scalar Wick contraction this is the last displayed clause of the supplied public conclusion. The exactness class is exact recovery of a limit; the subsequences and scaling families are retained witnesses, not part of the recovered quantity.

24.3 Trace measure patching and the ultraviolet comparison measure

Let 0, be the trace shot at imaginary-time height, centered at zero. Shot identities give and the two-point reading at heights () equals. Its diagonal momentum measure obeys

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dμΘ = e2p0dμ1.
(TR5)

Comparing any two heights through their sum gives equivalently for every Hence its damping is finite and has the regularized two-point characteristic function. Taking height2 proves TR3. Smearing and the established boundary limit give TR4 and joint-test Fourier representation. Cone support passes from shot measures; scalar Lorentz covariance and uniqueness identify every Lorentz pushforward with the same measure. The common polynomial shot-height bound gives its polynomial growth, detailed below.

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Shot-measure and undamping recovery.
Let P be the joint momentum spectral measure.
Under the same Hilbert unitary V , put PZ(B) = V P(B)V −1 and vZ
s = V vs. For every Borel set
B,
μZ
s (B) = ⟨vZ
s , PZ(B)vZ
s ⟩= ⟨vs, P(B)vs⟩= μs(B).

Functional calculus transports the heat semigroup. Comparing heights through therefore gives

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Interpret the second equality first on compact momentum sets, where weights are bounded, then
on all Borel sets by monotone exhaustion. This defines the possibly infinite undamped measure
without applying an inverse heat operator to a Hilbert vector. Its horizontal measure is exactly
μZ
Θ = μΘ, with
e−2sp0μZ
Θ = μZ
s ,
Z
e−2sp0dμZ
Θ = ∥vZ
s ∥2 < ∞.

Taking height2 recovers TR3’s time damping. Smearing and the same boundary map recover TR4 since Θ )Ω. Polarization and compact-test separation retain uniqueness; scalar Lorentz pushforwards therefore recover the same measure. Forward-cone support is unchanged. Temperate growth and joint Schwartz Fourier representation retain the polynomial shot-height and boundary dependencies developed below.

For the free contraction put, an orthogonal reflection, since multiplying its two rows and summing the common index gives. The curvature covariance is

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Curlabij(x) = D(x)(RaiRbj −RajRbi),
D(x) = 4 normal |x|−4.

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Orthogonality gives P
a,b,i,j Curl2
abij = 24D2: summing j, i first gives D2(2 −2δ2
ab), then summing
a, b gives 24D2. Scalar curvature-square has factor 1/4 in each field, two cross Wick contractions
and dim g colors. Its two-point constant is therefore (1/16)2 dim g 24D2 = 3 dim g/(π4|x|8) in the
source normalization.

Free contraction reader recovery. The horizontal free-label map retains the finite component indices, color contraction and scalar Wick reader. Its curvature covariance is the same) times the wedge of). For fixed, orthogonality gives

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! 

X
X
!2
= 2 −2δ2
ab.
X
X
j
R2
bj
i,j
(RaiRbj −RajRbi)2 = 2
i
R2
ai
−2
i
RaiRbi

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Summing the sixteen ordered pairs (a, b) gives 32 −8 = 24. The two scalar labels retain their
1/4 factors, the cross Wick contractions their multiplicity two and the color sum its dim g. The
recovered coefficient is therefore (1/4)2 2 24D(x)2 dim g, exactly 3 dim g/(π4|x|8) with the source
normalization. This transports the finite calculation used in TR2 with no new stress or Gaussian
normalization.
For ρ > 1 set Z(ρ) = ρ−8 log2 ρ and νρ = Z(ρ)(p 7→p/ρ)∗μΘ. These are positive Lorentz-
invariant cone measures, and TR2 yields
Z
e−sp0dνρ = Z(ρ)E(s/ρ) −→AGs−8
(s > 0),
(TR6)

since. Lorentz invariance gives the corresponding limit at every strictly future timelike damping vector. The candidate is

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dν∞(p) =
AG
1536π(p2
0 −|p|2)21{p0≥|p|}d4p.
(TR7)

Its Laplace normalization follows explicitly from

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4π
Z p0
0
(p2
0 −r2)2r2dr = 32π
105 p7
0,
Z ∞
0
e−sp0p7
0dp0 = 7!s−8.

The product is 1536. The cone-supported positive Laplace convergence argument below identifies this compact-test limit and supplies the stronger Schwartz-test passage. The lifted spectral source retains all shot measures and their exact semigroup relation. TR5 is an undamping reader independent of the chosen positive height, so it recovers the same trace measure and boundary pairings. The ultraviolet reader retains both the scaling pushforward and logarithmic weight, while TR7 records the normalized candidate with its spatial and temporal integral factors. No finite height measure is identified prematurely with the undamped one.

24.4 Polynomial trace bounds and Laplace-to-Schwartz continuity

On and) therefore give, uniformly for large

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νρ{p0 ≤R} ≤e
Z
e−p0/Rdνρ ≤CR8
(R ≥1).
(TR8)

The same holds for. On the coneis comparable to; dyadic shells and Schwartz decay give uniformly small tails. Compact-test convergence thus implies

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Z(ρ)
Z
φ(p/ρ)dμΘ(p) −→
Z
φ dν∞,
φ ∈S(R4).
(TR9)

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For fρ(x) = ρdf(ρx), change of variables gives bfρ,+(p) = bf+(p/ρ); the joint relative wave transform
has the same identity.
Hence TR9 gives the stated relative and joint Wightman limits.
The
polynomial growth and Laplace continuity assertions used in TR6–TR9 are proved next, followed
by the finite mass-shell obstruction; the subsequent stress correction and Ward calculation derive
the trace coefficient and its Wilson normalization.

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Ultraviolet pushforward and test recovery.
The spectral source retains the actual trace
measure, dilation and positive logarithmic weight.
Since μZ
Θ = μΘ, its ultraviolet measure is
identical on every Borel set:

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νZ
ρ (B) = Z(ρ)μZ
Θ(ρB) = νρ(B),
Z
φ dνZ
ρ = Z(ρ)
Z
φ(p/ρ) dμΘ(p).

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Thus the damping reader and all selected scaling parameters recover exactly. For fixed s > 0 and
sufficiently large ρ,
Z(ρ)
Z(ρ/s) = s−8

log ρ
log ρ −log s
2
−→s−8.

Combining with TR2 gives TR6 with no change of positive-distance quantifiers. The Lorentz action commutes with positive dilation and preserves the same cone, so timelike damping limits are transported by the identical measure action.

The comparison density retains its normalization independently of this limit. Substituting gives the spatial factor

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Z 1
1
3 −2
5 + 1
7

= 32π
105 p7
0.
0
(u2 −2u4 + u6) du = 4πp7
0
4πp7
0

Multiplication by 7! and) recovers. The horizontal scalar measure reader therefore keeps both the candidate density and its exact Laplace normalization. Its compact-test identification retains the positive Laplace continuity theorem proved next. For the uniform TR8 bound, the exact scale inequality is

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Z(ρ)
R8Z(ρR) =

log ρ
log ρ + log R
2
≤1
(R ≥1, ρ > 1).

Consequently the same large- threshold from TR2 controls all1. To make the Schwartz tail recovery explicit, choose 8 and set = sup. The uniform energy-slab bound controls each Euclidean dyadic shell and gives, for

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Z
∥p∥≥R
|φ(p)| dνρ(p) ≤28CML
1 −28−L R8−L.

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The same holds for ν∞. A fixed compact cutofffollowed by R →∞therefore transports compact-
test convergence to TR9 with uniform tails. Finally the normalized position dilation has its exact
Fourier readout bfρ,+(p) = bf+(p/ρ); the retained joint relative transform obeys the same change of

variables. This recovers the relative and joint distribution limits with the original wave convention and positive-measure dependencies intact. Explicitly, if a positive cone measure has

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μ{p : p0 ≤R} ≤ARk
(R ≥1),
(TR10)

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then its damped-vector bound supplies this with A = e2A0 when the squared height-R−1 norm is
at most A0Rk, since e−2p0/R ≥e−2 on the slab. Euclidean radius-R balls are in that slab. On
2j ≤1 + ∥p∥≤2j+1 the mass is at most A2k(j+1), so
Z
(1 + ∥p∥)−k−1dμ ≤2k+1A.
(TR11)

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Boundary overlap only enlarges the bound; this proves temperateness. Let positive μi, ν on the
closed cone C have finite Laplace transforms on its open dual C◦and satisfy
Z
e−u·pdμi −→
Z
e−u·pdν
(u ∈C◦).
(TR12)

Here damping uses the real Euclidean pairing. On the one-point compactification extends continuously by zero because u. These real functions multiply by . With constants they separate points: positivity separates infinity; perturbations of the time unit by small real vectors separate any two finite points. Stone–Weierstrass gives uniform density of their linear span. This is the real algebra approximation theorem of Stone [10], applied on the stated compactification with the point-separation and constant hypotheses just verified. Fix and damp all measures by. Their masses converge by TR12, and their integrals of

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constants and sv converge by the transforms at u and u + v. Uniform approximation and bounded
masses give convergence on the compactification’s continuous functions.
For ambient compact
continuous f, the function eu·pf on C, zero at infinity, is such a test and its damped integral is
R f dμi. This proves compact-test convergence, extending to complex tests by real and imaginary
parts. For common TR10 bounds on all measures and the limit, put q = 2k+1A, and choose B with
|f(p)| ≤B(1 + ∥p∥)−k−2. Smooth χR equal to one on radius R and zero beyond R + 1, 0 ≤χR ≤1,
gives
Z
Z
|f −χRf|dμi ≤Bq/R,
|f −χRf|dν ≤Bq/R.

First choose for the tails, then apply compact convergence to This proves TR9 with uniform tails. The same density argument identifies a positive cone measure uniquely from all its future Laplace transforms; no signed-measure convergence is asserted. The lifted measure source retains positivity, cone support, the common polynomial budget and damping family. Its compact reader is recovered by bounded-mass approximation after damping, and its Schwartz reader by the explicit tail. Keeping the normalized dilation and joint relative transform transports the same ultraviolet measure to the full tested two-point distributions without exchanging uncontrolled limits.

Positive Laplace continuity and Schwartz recovery. All measure operations here are readouts of the retained positive cone measure, so the height bound and its spectral integral agree under the horizontal Hilbert realization. Explicitly

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μ{p0 ≤R} ≤e2
Z
e−2p0/Rdμ = e2∥v1/R∥2 ≤e2A0Rk.